cp's OEIS Frontend

A324085 Digits of one of the four 3-adic integers 3^(1/4) that is congruent to 2 mod 13.

%I A324085 #9 Sep 07 2019 19:09:26
%S A324085 2,2,7,1,12,6,12,4,8,6,1,10,4,6,7,8,10,1,12,9,9,7,0,12,3,6,4,5,11,12,
%T A324085 3,11,9,5,8,4,4,2,7,4,11,8,4,10,1,0,2,1,4,3,11,7,3,6,3,2,6,7,3,6,1,0,
%U A324085 3,0,11,8,11,6,11,0,3,5,4,7,9,10,12,6,11,5,1
%N A324085 Digits of one of the four 3-adic integers 3^(1/4) that is congruent to 2 mod 13.
%C A324085 One of the two square roots of A322087, where an A-number represents a 13-adic number. The other square root is A324153.
%C A324085 For k not divisible by 13, k is a fourth power in 13-adic field if and only if k == 1, 3, 9 (mod 13). If k is a fourth power in 13-adic field, then k has exactly 4 fourth-power roots.
%H A324085 Wikipedia, <a href="https://en.wikipedia.org/wiki/P-adic_number">p-adic number</a>
%F A324085 Equals A324086*A286838 = A324087*A286839.
%F A324085 a(n) = (A324077(n+1) - A324077(n))/13^n.
%F A324085 For n > 0, a(n) = 12 - A324153(n).
%e A324085 The unique number k in [1, 13^3] and congruent to 2 modulo 13 such that k^4 - 3 is divisible by 13^3 is k = 1211 = (722)_13, so the first three terms are 2, 7 and 7.
%o A324085 (PARI) a(n) = lift(sqrtn(3+O(13^(n+1)), 4) * sqrt(-1+O(13^(n+1))))\13^n
%Y A324085 Cf. A286838, A286839, A322087, A324077, A324082, A324083, A324084, A324086, A324087, A324153.
%K A324085 nonn,base
%O A324085 0,1
%A A324085 _Jianing Song_, Sep 01 2019